Fundamentals and Applications of Complex Analysis

This book is intended to serve as a text for both beginning and second courses in complex analysis of functions of one complex variable. The material that is appropriate for more advanced study is developed from elementary material. The concepts are illustrated with large numbers of examples, many of which involve problems students encounter in other courses. For example, students who have taken an introductory physics course will have encountered analysis of simple AC circuits. This text revisits such analysis using complex numbers. Cauchy's residue theorem is used to evaluate many types of definite integrals that students are introduced to in the beginning calculus sequence. Methods of conformal mapping are used to solve problems in electrostatics. The book contains material that is not considered in other popular complex analysis texts. For example, one chapter is devoted to an analysis of multivalued functions, with applications to the evaluation of certain types of integrals. Another chapter deals with the singularity structure of functions that are defined by integrals which cannot be evaluated in terms of elementary functions. A third chapter develops dispersion relations, which are mathematical tools for determining a complete function from a knowledge of just the real part, or just the imaginary part of the function.

Table of Contents:

1. Introduction. 2. Complex Numbers. 3. Complex Variables. 4. Series, Limits and Residues. 5. Evaluation of Integrals. 6. Multivalued Functions, Branch Points and Cuts. 7. Singularities of Functions Defined by Integrals. 8. Conformal Mapping. 9. Dispersion Relations. Appendix 1: Derivation of Green's Theorem. Appendix 2: Derivation of the Geometric Series. Appendix 3: Evaluation of an Integral. Appendix 4: Transformation of Laplace's Equation. Appendix 5: Transformation of Boundary Conditions. Index.